Virtually homo-uniserial modules and rings

Abstract

We study the class of virtually homo-uniserial modules and rings as a nontrivial generalization of homo-uniserial modules and rings. An R-module M is virtually homo-uniserial if, for any finitely generated submodules the factor modules and are virtually simple and isomorphic (an R-module M is virtually simple if, and for every nonzero submodule N of M). Also, an R-module M is called virtually homo-serial if it is a direct sum of virtually homo-uniserial modules. We obtain that every left R-module is virtually homo-serial if and only if R is an Artinian principal ideal ring. Also, it is shown that over a commutative ring R, every finitely generated R-module is virtually homo-serial if and only if R is a finite direct product of almost maximal uniserial rings and principal ideal domains with zero Jacobson radical. Finally, we obtain some structure theorems for commutative (Noetherian) rings whose every proper ideal is virtually (homo-)serial.